Both names share the ultimate root of Latin "felix" meaning "lucky". The names are related in origin but were not strictly made as masculine / feminine variants of each other.
As far as I'm aware
Felix is not known as a nickname for
Felicity, but that doesn't discount being able to use it by any means. Nicknames and diminutives can take many forms, in fact the name
Felix itself is derived from a nickname for a lucky person.
! - Edited to answer second part of question. Possible nicknames for
Felicity are:
FlickFliss
Feliz
Fee
Feli
Fel
Fifi Lissie
This message was edited 3/22/2017, 7:29 AM